The inclusion of initial spatial clustering of PBHs changes the story. The initial clustering of PBHs forms the high density PBHs regions in radiation-domination epoch. As we have shown in [1]}, the initial clustering of PBHs can be constructed in multi-stream inflation [2]}, [3]}, [4]}, [5]}. During multi-stream inflation, inflaton travels along two inflation trajectories, inflation potential on one trajectories can produce PBHs, the others can not produce PBHs. In the PBHs clustering model [1]}, the PBHs clustering regions are denoted by \(\beta \) , which is the volume fraction of PBHs regions in the observed universe. The redshift of matter-radiation equality in PBHs rich regions can be calculated by increasing the PBHs fraction in dark matter in PBHs rich region \(f_{\mathrm {PBH}} \rightarrow f_{\mathrm {PBH}}/\beta \)  as following
\(1 + z_{\mathrm {PBH}, \mathrm {eq}} = (1 - f_{\mathrm {PBH}} + \frac{f_{\mathrm {PBH}}}{\beta }) (1 + z_{\mathrm {eq}})~.\) 
